# Divergence issue with my monte carlo pricer…

I am trying to implement a vanilla European option pricer with Monte Carlo and compare its result to the BS analytical formula's result.

I noticed that as I increase (from 1 million to 10 millions) the number of simulations, the MC result starts to diverge.

Note that I deliberately only use only one variance reduction technique: antithetic variables. I was hoping that by merely increasing the number of simulations, I would manage to increase the precision.

Can anyone please give me clues or pointers as to why my result diverges?

Included below is the C# code for the pricer:

using System;
using MathNet.Numerics.Distributions;
using MathNet.Numerics.Random;

namespace MonteCarlo
{
class VanillaEuropeanCallMonteCarlo
{
static void Main(string[] args)
{
const int NUM_SIMULATIONS = 10000000;
const decimal strike = 50m;
const decimal initialStockPrice = 52m;
const decimal volatility = 0.2m;
const decimal riskFreeRate = 0.05m;
const decimal maturity = 0.5m;
Normal n = new Normal();
n.RandomSource = new MersenneTwister();

VanillaEuropeanCallMonteCarlo vanillaCallMonteCarlo = new VanillaEuropeanCallMonteCarlo();

for (int i = 0; i < simulations.Length; i++)
{
simulations[i] = new Task<decimal>(() => vanillaCallMonteCarlo.RunMonteCarloSimulation(strike, initialStockPrice, volatility, riskFreeRate, maturity, n));
simulations[i].Start();
}

decimal total = 0m;

for (int i = 0; i < simulations.Length; i++)
{
total += simulations[i].Result;
}

decimal callPrice = (decimal)(Math.Exp((double)(-riskFreeRate * maturity)) * (double)total / (NUM_SIMULATIONS * 2));

Console.WriteLine("Call Price: " + callPrice);
Console.WriteLine("Difference: " + Math.Abs(callPrice - 4.744741008m));
}

decimal RunMonteCarloSimulation(decimal strike, decimal initialStockPrice, decimal volatility, decimal riskFreeRate, decimal maturity, Normal n)
{
decimal randGaussian = (decimal)n.Sample();
decimal endStockPriceA = initialStockPrice * (decimal)Math.Exp((double)((riskFreeRate - (decimal)(0.5 * Math.Pow((double)volatility, 2))) * maturity + volatility * (decimal)Math.Sqrt((double)maturity) * randGaussian));
decimal endStockPriceB = initialStockPrice * (decimal)Math.Exp((double)((riskFreeRate - (decimal)(0.5 * Math.Pow((double)volatility, 2))) * maturity + volatility * (decimal)Math.Sqrt((double)maturity) * (-randGaussian)));
decimal sumPayoffs = (decimal)(Math.Max(0, endStockPriceA - strike) + Math.Max(0, endStockPriceB - strike));
return sumPayoffs;
}
}
}

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This is essentially the same question as your previous question and the issue is still the same: variability just does not go away just because you use 100 million draws once. Compare the distribution of results of $N$ Monte Carlo simulations at $n_1 = 1,000,000$ with those for $n_2 = 10,000,000$. You will see a reduction but that does not imply that every single run get a tighter answer.